Anomalous Silica Colloid Stability and Gel-Layer-Mediated

Article pubs.acs.org/Langmuir

Anomalous Silica Colloid Stability and Gel Layer Mediated Interactions Julie L. Bitter,† Gregg A. Duncan,‡ Daniel J. Beltran-Villegas,‡ D. Howard Fairbrother,† and Michael A. Bevan*,‡ †

Department of Chemistry and ‡Department of Chemical & Biomolecular Engineering, Johns Hopkins University, Baltimore, Maryland 21218, United States S Supporting Information *

ABSTRACT: Total internal reflection microscopy (TIRM) is used to measure SiO2 colloid ensembles over a glass microscope slide to simultaneously obtain interactions and stability as a function of pH (4−10) and NaCl concentration (0.1−100 mM). Analysis of SiO2 colloid Brownian height excursions yields kTscale potential energy vs separation profiles, U(h), and diffusivity vs separation profiles, D(h), and determines whether particles are levitated or irreversibly deposited (i.e., stable). By including an impermeable SiO2 “gel layer” when fitting van der Waals, electrostatic, and steric potentials to measured net potentials, gel layers are estimated to be ∼10 nm thick and display an ionic strength collapse. The D(h) results indicate consistent surface separation scales for potential energy profiles and hydrodynamic interactions. Our measurements and model indicate how SiO2 gel layers influence van der Waals (e.g., dielectric properties), electrostatics (e.g., shear plane), and steric (e.g., layer thickness) potentials to understand the anomalous high ionic strength and high pH stability of SiO2 colloids.

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direct measurements5−8 do not conclusively support either mechanism or a quantitative potential model. It is also not clear that the role of silica gel layers has been treated self-consistently in terms of their effects on all interactions including van der Waals, electrostatic, and steric interactions. In this work, we simultaneously measure the interactions and stability of silica colloids over a glass microscope slide as a function of pH (4−10) and ionic strength (0.1−100 mM NaCl). TIRM is used to nonintrusively measure weak kT-scale interactions between a glass microscope slide and an ensemble of silica colloids14 by analyzing their Brownian height excursions, which also reveals whether they are irreversibly deposited or levitated (i.e., stable). This has the advantage that separation-dependent interactions are obtained simultaneously with measurements of stability, so the two can be unambiguously linked in the same material system. We also simultaneously obtain potential energy vs separation, U(h), and diffusivity vs separation, D(h), profiles by fitting the Smoluchowski equation coefficients to the measured particle dynamic trajectories. The D(h) trajectories yield additional information about particle−wall separation and fluid mechanics important to interpretation of electrostatic and steric interactions. As such, the present study provides new measurements and models of silica gel-layer-mediated interactions that lead to anomalous silica colloid stability.

INTRODUCTION Silica is ubiquitous. It makes up 60% of the earth’s crust, and silicates make up 90% of all minerals. It is present in amorphous and crystalline forms important for everyday use, optics, and microelectronics. It is present in food, pharmaceuticals, and organisms. As such, understanding the chemical and physical properties of bulk silica, silica surfaces, and colloidal silica is crucial to numerous applications. A great deal of silica chemistry is well understood and catalogued in the comprehensive book by Iler.1 However, the stability of colloidal silica against aggregation at high ionic strengths and high pHs is often referred to as “anomalous”2 because it is not well described by the Derjaguin−Landau−Verwey−Overbeck (DLVO) theory.3,4 Historical reviews of possible stabilizing mechanisms that might account for anomalous silica colloid stability are contained within representative papers on the topic.2,5−8 Direct measurements of force vs distance curves with the surface forces apparatus and the atomic force microscope indicate a short-range repulsion.5−8 While this measured repulsion appears sufficient to account for anomalous colloidal stability, its physical origin remains an open question. Two mechanisms suggested in the literature include structural forces due to interfacial water9 or steric interactions between silica gel layers.10 The water structuring mechanism does not appear to be unique to silica. The presence of silica gel layers is supported by measurements of adhesion, friction, contact angle,5 and surface density profiles via scattering/spectroscopic methods.11−13 Despite some evidence in favor of silica gel layers, © 2013 American Chemical Society

Received: April 29, 2013 Revised: June 17, 2013 Published: June 18, 2013 8835

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where the Δ terms include the frequency-dependent dielectric properties of the particle (1), wall (2), and medium (3) and the remainder of the terms are defined in previous papers.18,19 The prime (′) next to the summation indicates that the first term (n = 0) is multiplied by 1/2(1 + 2κl)exp(−2κl) to account for screening of the zero-frequency contribution. The interaction between surfaces coated with macromolecular layers depends on the free energy change of layers under compression.21−23 For a layer with a brush architecture on a planar surface with an uncompressed thickness, δ0, and free energy per area, f 0, the compressed free energy per area, f(δ), for 1/2 < δ/δ0 < 1 can be captured accurately by24

THEORY Potential Energy Profiles. By measuring a statistically significant number of height excursions, h, of a spherical particle above a planar wall surface, a normalized equilibrium height histogram, p(h), can be related to net separation-dependent interaction potential, U(h), via Boltzmann’s equation as p(h) = exp[−U (h)/kT ]

(1)

where k is Boltzmann’s constant and T is absolute temperature. Equation 1 can be inverted to obtain a measurement of U(h) from measured p(h) data as

U (h) = −kT ln[p(h)]

(2)

f (δ)/f0 = 1 + ΓBexp[−γB(δ /δ0)]

Theoretical models of U(h) can be computed from superposition of contributing potentials as U (h) = UG(h) + UE(h) + UV(h) + US(h)

where ΓB and γB are dimensionless constants specific to the brush architecture.24 Using this expression in the Derjaguin approximation, the potential between two symmetric macromolecular brush layers as can be obtained as24

(3)

where the subscripts refer to the gravitational (G), electrostatic (E), van der Waals (V), and steric (S) interactions. The gravitational potential is associated with a body force, whereas the other potentials are associated with surface forces. Electrostatic and van der Waals potentials were considered in the original DLVO theory.3,4 The gravitational potential energy of each particle depends on its height, h, of the particle above the wall, multiplied by its buoyant weight, G, as given by UG(h) = Gh = mgh = (4/3)πa3(ρp − ρf )gh

US,B(h) = 16πaf0 δ0(ΓB/γ )exp[−(hγB/2δ0)]

US(h) = Γ exp( −γh)

where m is buoyant mass, g is acceleration due to gravity, and ρp and ρf are particle and fluid densities. The interaction between electrostatic double layers on two plates (from superposition, nonlinear Poisson−Boltzmann equation, 1:1 monovalent electrolyte)15 can be used in conjunction with the Derjaguin approximation to give the particle−wall potential as16 (5)

⎛ kT ⎞2 ⎛ eψ ⎞ ⎛ eψ ⎞ B = 64πεa⎜ ⎟ tanh⎜ 1 ⎟tanh⎜ 2 ⎟ ⎝ 4kT ⎠ ⎝ 4kT ⎠ ⎝ e ⎠

(6)

⎡ e 2N κ=⎢ A ⎢⎣ εkT

(7)

UV(h) = (a /6)

∫h

D(h) = D0f (h)

D0 = (8)

∞

∫r

∞

kT 6πηa

x{[1 − Δ13Δ23e−x]

n

−x

+ ln[1 − Δ̅ 13Δ̅ 23e ]}dx

(15)

where η is the fluid medium viscosity and f(h) accounts for particle−wall hydrodynamic interactions from Brenner,26 which is accurately captured by the simple expression27

where A(l) is the Hamaker function given by19,20 3 A(l) = − kT ∑ ′ 2 n=0

(14)

where D0 is the Stokes−Einstein coefficient of an unbounded spherical particle given by

2

−A(l)/l dl

(13)

which reduces to Boltzmann’s equation in the long-time limit as equilibrium is approached. In previous work, we reported nonequilibrium analysis of colloidal trajectories to obtain U(h) and D(h). Measured D(h) are modeled using

where κ is the inverse Debye length, ε is the solvent dielectric constant, e is the elemental charge, ψ1 and ψ2 are surface potentials, NA is Avogadro’s number, Ci is electrolyte molarity, and zi is ion valence. van der Waals attraction between two plates as predicted from the Lifshitz theory17 (which includes retardation and screening effects) can be used in conjunction with the Derjaguin approximation to give the particle−wall potential as18 ∞

(12)

is broadly applicable (by lumping unknown constants together), particularly when the uncompressed layer properties and architecture are not well characterized (which has also been shown for asymmetric interactions between layers of different properties24). Diffusivity Profiles. In contrast to measuring the equilibrium probability p(h) to obtain U(h) via eq 2, measurements of the time-dependent probability, p(h,t), can be used to obtain both U(h) and the separation-dependent diffusivity, D(h), as described by the Smoluchowski equation25 ⎡ ∂p(h , t ) ∂p(h , t ) p(h , t ) dU (h) ⎤ ∂ = D(h)⎢ + ⎥ ⎣ ∂h ∂t ∂h kT dh ⎦

⎤1/2 2 ⎥ ∑ (zi) Ci ⎥⎦ i

(11)

For different interfacial macromolecular architectures with different decaying density profiles at their periphery, different values of Γ and γ can be used in eq 10. Because eq 11 can be used to accurately model adsorbed macromolecular layer repulsion for a broad range of δ0, f 0, Γ, and γ, a general repulsive steric potential of the form

(4)

UE(h) = B exp[−κh]

(10)

f (h) =

(9) 8836

6h2 + 2ah 6h + 9ah + 2a 2 2

(16)

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MATERIALS AND METHODS

Colloids and Surfaces. Hydrochloric acid, potassium hydroxide, sodium chloride (all from Fisher Scientific), and colloidal SiO2 (nominal diameter of 2.34 μm, Bangs Laboratories) were used as purchased and without further purification. Nonporous amorphous SiO2 colloids were synthesized by a precipitation method using pure reagents with minimal trace elements. Glass microscope slides (Fisherbrand Plain Microscope Slides) had a manufacturer reported density of 2.48 g/cm3 and a soda lime composition (approximately 72% silicon dioxide, 14% sodium oxide, 6% calcium oxide, 4% magnesium oxide, 1% aluminum oxide, 1% potassium oxide, 10 mM in Figure 2A, we also use D(h) data to confirm the validity of the separation scale inferred from the conservative forces and confirm some assumptions about the fluid flow in the presence of gel layers. Before discussing the potential fits that include steric contributions in Figure 2A, we first discuss several conceptual issues related to including a gel layer in a manner that consistently treats electrostatic and van der Waals potentials. Role of “Gel” Layer in van der Waals, Electrostatic, and Steric Potentials. The primary conceptual issue to address for computing net potentials in the presence of silica gel layers is the reference separation for each potential. Four illustrative cases are depicted in Figure 3 showing (1) no gel layer with UE and UV on a scale, h, between the H2O/SiO2 interfaces, (2) a gel layer composed of nearly pure SiO2 with UE and UV on the h scale and a steric interaction, US, on a separation scale, L, between the SiO2 gel/bulk interfaces, (3) a gel layer with mostly water properties that is permeable to fluid flow and has charge on the SiO2 gel/bulk interfaces (this suggests UE, UV, and US should all be on the L scale), and (4) a gel layer with mostly water properties that is impermeable to fluid flow and has a no-slip surface/potential originating on the H2O/SiO2 gel interfaces (this suggests UV and US should be on the L scale and UE should be on the h scale). This list is not exhaustive, but these are the four physically meaningful cases that bound other cases including gel layer dielectric properties intermediate to pure H2O/pure SiO2 and/or charge distributed (and a shear plane) between the H2O/SiO2 and the SiO2 gel/ bulk interfaces. These cases can be compared and contrasted to decide on an appropriate model for the net potentials in Figure 2A. Case 1 is the standard model for the DLVO theory. Case 2 illustrates a gel layer that has no stabilizing effect. In particular, if the gel layer is composed almost entirely of SiO2, the van der Waals attraction will still be as strong on the h scale as in case 1.40−42 However, since steric repulsion between the gel layers is not generated until h = 0, the strong van der Waals attraction for h > 0 would cause irreversible surface adhesion. In fact, cases 1 and 2 have the same van der Waals and electrostatic interactions, but the repulsion at contact is weaker (soft steric repulsion instead of hard wall repulsion). Clearly a different mechanism is required to produce stabilization by a silica gel layer. Cases 3 and 4 illustrate gel layers with a predominantly water composition and therefore water dielectric properties. This effectively weakens van der Waals on the h scale by moving UV to the L scale in the limit of a purely water layer. The difference between cases 3 and 4 is whether the electrostatic potential originates at the SiO2 gel/bulk interfaces (case 3) or the H2O/ SiO2 gel interfaces (case 4). These cases are limits of intermediate cases that depend on whether the layer is a polyelectrolyte, how charge is spatially distributed within the layer, and whether ions are mobile within the layer.10,43 Both of these cases can produce net potentials that correspond to stable particles and can be fit to the data in Figure 2A. In short, case 4 has more electrostatic repulsion and as a result has a smaller steric contribution, whereas case 3 has less electrostatic repulsion and therefore requires a greater steric contribution. The key effect in cases 3 and 4 compared to case 2 is that van der Waals attraction is weaker due to the gel layer.

Figure 3. Schematics and predicted potentials (for pH = 10, [NaCl] = 80 mM in Figure 2A) based on various cases for including SiO2 gel layers. In the schematics and predictions, h is the separation between the outer edges of the SiO2 gel layers (i.e., the H2O/SiO2 gel interfaces) and L is the separation between the inner edges of the SiO2 gel layers (i.e., the SiO2 gel/bulk interface). See text for detailed explanation of each case, but in brief (top-to-bottom) (1) the typical configuration with no gel layers considered in the DLVO theory, (2) gel layers of mostly SiO2 composition, (3) gel layers of mostly H2O composition that are permeable to fluid flow, (4) gel layers of mostly H2O composition that are impermeable to fluid flow. Potentials are color coded as electrostatics (red), van der Waals (blue), steric (yellow), and net (green). 8839

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Fitting DLVO and Steric Interactions in the Presence of a “Gel” Layer. Based on the discussion of the different cases in Figure 3, the measured profiles for [NaCl] > 10 mM in Figure 2A are fit using models based on cases 3 and 4. As in the purely DLVO fits for [NaCl] < 5 mM, the same ionic strength and pH-dependent κ from eq 7 and the ψ model in the Supporting Information27,28 are used in the DLVO potentials for [NaCl] > 10 mM. For the steric potential in eq 12, a prefactor of Γ = 100kT is assumed and the steric inverse decay length, γ, becomes the sole adjustable parameter in the net potential. While Γ = 100kT is somewhat arbitrary, it is of the correct order of magnitude based on the few cases where the steric prefactor has been estimated22 and based on what is necessary to generate the observed stability. We decided to fix Γ = 100kT for several reasons including (1) a greater sensitivity of the fit to the decay length than the prefactor when both parameters are varied, (2) some uncertainty in the strong, short-range repulsion due to noise, which affects estimates of the intercept, and (3) the prefactor in steric interactions cannot generally be predicted a priori based on independent parameters.22 On the basis of this prefactor, the gel layer thickness, Δ, can be estimated as 2Δ = L − h = 5γ−1, which corresponds to a decay from 100kT to ∼0.5kT based on the properties of the exponential function. As a result, the difference between cases 3 and 4 is simply whether all potentials are on the same separation scale (case 3) or whether the electrostatic potential is shifted outward by 5γ−1 (case 4). We return to a discussion of the validity of the assumed prefactor after reporting and discussing the fit steric decay lengths. Figure 4 reports the inferred decay lengths, γ−1, and gel thicknesses, Δ = 0.5(L − h) = 2.5γ−1, as a function of ionic

location at the most probable separation, hm. This approach is necessary since the actual repulsive decay corresponds to a strong force approaching the noise limit of the TIRM method,44,45 which limits the accuracy of the measured repulsive decay length. In short, noise softens strong forces (i.e., large energy changes over small distances) at short separations and high ionic strengths, so that matching the potential energy minimum well depth and location is a better measure of the net repulsion than the decay length. When considering the absolute values of the layer dimensions inferred in Figure 4, it is important to recall that cases 3 and 4 bound some limiting physical models. For example, distributing charge anywhere in between the SiO2 gel/ bulk interfaces (case 3) or the H2O/SiO2 gel interfaces (case 4) would produce gel layer estimates in between the two curves shown in Figure 4. If the layers contain a higher concentration of SiO2 than the nearly pure H2O layers in cases 3 and 4 then the two cases in Figure 4 represent lower bounds where the layer thickness would diverge to infinity as the layers approach pure SiO2 (as in case 2 where the gel layer is not capable of generating a stabilizing repulsion beyond the range of van der Waals attraction). One way to overcome this problem is to also include surface roughness in addition to the gel layer, which will weaken van der Waals18 and still allow gel layers without pure water properties.46,47 Do Inferred Gel Layer Properties Make Sense? The inferred gel layers in Figure 4 from the measured potentials in Figure 2 display the expected trend by showing a decreasing thickness vs increasing ionic strength. This behavior is consistent with the gel behaving as a polyelectrolyte where screening of electrostatic repulsion within the layers allows for a dimensional collapse.48,49 It is also expected that this collapse will not occur until high ionic strengths when the Debye length is on the order of the separation of charges within the gel layer. This dimensional collapse will expel water from the gel layer and enrich it in pure SiO2 properties, which is reminiscent of solvent-quality-mediated collapse of adsorbed polymer layers.42,46,50,51 Such a collapse will reduce stability by both decreasing steric repulsion and increasing the van der Waals attraction due to changing layer dielectric properties.42 The observed stability behavior also has the character of a solvent quality mediated collapse of a repulsive steric interaction. Specifically, an attractive energy minimum evolves beyond the range of an infinitely repulsive steric barrier, which progressively increases bond lifetimes with an exponential dependence on well depth.24,43 When the attractive well is deep enough to produce most probable bond lifetimes longer than the observation time, then the particle appears to be irreversibly deposited. This type of destabilization mechanism contrasts the typical mechanism for only DLVO interactions, where the height of an energy barrier determines the probability of forming an irreversible bound state involving strong van der Waals attraction at contact. Our results appear to display stability behavior consistent with a decreasing range of steric repulsion due to collapsing impenetrable gel layers. The values of the inferred thicknesses are larger than the ∼2 nm estimates from mechanical force measurements (e.g., SFA,5 AFM6) but comparable to gel thicknesses obtained from surface spectroscopic/scattering methods (e.g., nuclear resonance profiling,11 neutron, X-ray reflectivity12,13). One way to reconcile the differences between layer thicknesses inferred from SFA and AFM force profiles and the TIRM potential energy profiles in Figure 2A is the much higher sensitivity to

Figure 4. Steric decay length, γ−1 (left), and twice the single gel layer thickness, 2Δ (right), vs [NaCl]/mM at pH = 10 from fits in Figure 2A based on models for cases 3 (red triangles) and 4 (blue circles) in Figure 3.

strength from the fits to the pH = 10 profiles in Figure 2. Data corresponding to fits based on case 3 indicate decay lengths of γ−1 ≈ 4−7 nm and thicknesses of Δ ≈ 10−17 nm, whereas fits based on case 4 give γ−1 ≈ 2−4 nm and Δ ≈ 5−10 nm. It should be noted that these fit parameters are obtained by generating a repulsion that results in the correct attractive well depth, which then determines the potential energy minimum 8840

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small energies (and forces) with TIRM.37 In particular, very weak steric interactions between silica gel layers could go undetected with mechanical methods until they generate ≥10 pN of repulsion. In contrast, TIRM is capable of detecting the very onset of silica gel layer compression on the kT energy scale and fN force scale (e.g., 1 kT/100 nm ≈ 10 fN). This sensitivity can be expected to produce thicker layer estimates, and because TIRM is not mechanically limited, it might produce estimates closer to nonintrusive spectroscopic/scattering methods. In short, the 5−17 nm SiO2 gel layers are reasonable based on literature neutron and X-ray measurements.12,13 It is also useful to consider the D(h) data and fits in Figure 2B. The D(h) fits serve the purpose of confirming that steric potentials occur on separation scales consistent with hydrodynamically measured surface separation. In particular, estimates of hm from non-DLVO fits to U(h) data in Figure 2A for both cases 3 and 4 agree with hm estimates from the D(h) fits in Figure 2B within the uncertainty of the measurements (see Figure S1, Supporting Information). The pH = 10, [NaCl] = 80 mM potentials in Figure 3 can be used to illustrate this point; the net potential in Figure 3C for case 3 has Lm = 13 nm, whereas the net potential in Figure 3D for case 4 has hm = 8 nm, which are both within the uncertainty of hm = 7 nm from the D(h) fit in Figure 2B (it should be noted that Lm and hm are used in this example since L is the hydrodynamic separation scale in case 3 and h is the hydrodynamic separation scale in case 4). As a result, steric potentials with Γ = 100 kT and layer thicknesses of Δ = 2.5γ−1 produce net U(h) profiles on the same separation scale as the D(h) data and fits. However, the D(h) fits do not resolve difference between cases 3 and 4. Of cases 3 and 4 presented in Figure 3, case 4 is more likely for several reasons. Because the electrostatic repulsion is longer range in case 4, the silica gel layer thicknesses in case 4 are thinner and closer to the estimates from spectroscopic/ scattering methods. On the basis of previous measurements of adsorbed polymers,46 it is more likely the gel layers are impermeable to flow (case 4). The electrostatic potential appears more likely to originate from a surface potential at the outer edge of the silica gel layer (case 4). Potentials and Stability vs Ionic Strength and pH. To understand how the silica gel layer influences stability as a function of both pH and ionic strength, Figure 5 reports results that summarize both stability and potential energy profile measurements. In addition to the pH = 10 results already discussed in Figures 2−4, ionic-strength-dependent results are shown for pHs of 7, 5.5, and 4. The points show several states indicating whether (1) all particles were robustly levitated throughout the entire observation time and had potentials captured by DLVO theory (green circles), (2) all particles were robustly levitated and had potentials that required DVLO theory + a steric repulsion (green triangles), (3) some particles became irreversibly deposited during the observation time and the particles that remained levitated had potentials that required DVLO theory + a steric repulsion (yellow inverted triangles), or (4) all particles were irreversibly deposited during the observation time (red squares). The measured ionicstrength-dependent potentials for each pH are included in Figures S4−S6, Supporting Information, with fit parameters reported in Table 2. The agreement between measured potentials and theoretical fits for other pHs in Figures S4− S6, Supporting Information, is similar to the agreement observed for the pH = 10 data in Figure 2A.

Figure 5. Summary of whether DLVO theory fit measured potentials and the degree of particle stability vs solution pH and [NaCl]. Points indicate (1) robust levitation, accurately modeled by DLVO theory (green circles), (2) robust levitation, modeled by DLVO + steric repulsion (green triangles), (3) slow deposition of particles, levitated particles are modeled by DLVO + steric repulsion (yellow inverted triangles), and (4) irreversible deposition (red squares).

At each pH, there is a clear progression with increasing ionic strength through states 1−4 described in the previous paragraph. The ionic strength dependence is different at each pH showing a more compressed transition through states 1−4 at lower pHs. For example, at pH = 10 the transition from stable particles described by DLVO theory to irreversibly deposited particles occurs between [NaCl] = 5 and 100 mM, whereas the same transition occurs at pH = 4 between [NaCl] = 0.1 and 10 mM. At each pH, the potentials and stability are well described by DLVO theory at low ionic strengths, but an additional repulsion, presumably due to steric gel layer interactions, is required to fit the measured potentials and capture the stability at high ionic strengths. The agreement between measured low ionic strength potentials with DLVO theory is easy to understand at all pHs; the long-range electrostatic repulsion does not allow particle−wall separations to become small enough to observe a steric repulsion between silica gel layers in contact. However, as the ionic strength is increased at each pH, steric repulsion is required to capture the observed repulsion, lack of attraction, and stability at higher ionic strengths. The fits to measured potentials at all pHs and ionic strengths (Figure 2A, Figures S4−S6, Supporting Information) have no adjustable parameters in the electrostatic and van der Waals contributions using κ from eq 7 and the ψ model in the Supporting Information.27,28 The different trends at each pH in Figure 5 are accounted for by the ionic strength and pH dependence of the van der Waals and electrostatic potentials, which suggests the silica gel layer repulsion that is relatively insensitive to pH. This is perhaps most readily illustrated by noting the steric decay length vs solution ionic strength is essentially the same for each pH within the limits of uncertainty of the fit points (see γ−1 data in Table 2). Because intramolecular electrostatic repulsion within a polyelectrolyte brush determines its degree of swelling,48 decreasing such interactions either by screening at elevated ionic strengths or by reducing the total charge via pH8841

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Table 2. Experimental Parameters for Each pH and Ionic Strength Condition Examineda pH

2a (μm)

NaCl (mM)

κ−1 (nm)

pH

−Ψ(mV)

γ−1(#3) (nm)

hm-U (nm)

γ−1 (#4) (nm)

hm-U (nm)

hm-D (nm)

10

2.17 2.15 2.15 2.16 2.13 2.17 2.13 2.15 2.15 × 2.13 2.22 2.20 2.15 2.13 2.08 × 2.22 2.10 2.08 2.07 2.02 2.08 × 2.18 2.13 2.03 ×

0.11 2.1 5.2 9.9 16.1 21.1 40.6 60.8 84.2 111 0.05 2.2 5.0 11.3 17.2 22.8 34.1 0.04 2.1 5.1 10.3 16.3 20.9 31.8 0.13 2.0 4.5 11.1

30.0 6.6 4.2 3.0 2.5 2.1 1.5 1.2 1.1 0.96 44.2 6.8 4.3 3.0 2.5 2.1 1.8 51.2 6.8 4.3 3.0 2.5 2.1 1.8 29.9 6.6 4.2 3.0

10.02 10.01 10.00 9.93 10.01 9.91 9.88 9.92 9.77 9.94 7.01 7.10 6.99 7.05 7.00 7.01 7.08 5.52 5.48 5.50 5.51 5.55 5.44 5.46 3.98 4.00 3.97 3.98

120 100 83 61 44 36 26 25 24 24 94 85 70 49 36 30 25 70 64 54 40 30 24 19 30 29 25 15

7.2 6.3 5.9 4.7 4.3 4.1 × 7.0 6.7 6.4 5.5 × 8.2 8.5 6.8 7.6 × 10.0 9.3 ×

320.5 76.4 48.1 39.2 35.1 29.7 17.6 14.1 13.4 × 443.6 76.7 47.6 37.7 35.4 25.1 × 480.0 73.6 53.0 47.8 34.8 43.3 × 245.7 67.7 57.6 ×

4.2 3.4 3.5 2.33 2.1 2.0 × 3.1 2.6 3.4 2.7 × 2.7 7.0 4.8 6.0 × 3.0 4.0 ×

320.5 76.4 48.1 33.4 25.9 21.5 11.6 9.1 7.5 × 443.6 76.7 47.8 30.6 23.8 18.4 × 480.0 73.6 47 30.3 21.6 20.3 × 245.7 62.6 39.7 ×

300 80 50 45 30 25 15 7 7 ×

7

5.5

4

The column labeled “γ−1 (#3)” is the steric decay length from a net potential fit based on case 3 in Figure 3 and “γ−1 (#4)” is the steric decay length from a net potential fit based on case 4. Column labeled as hm-U is the most probable particle−wall separations obtained from potential energy profile fits, and hm-D is the most probable height from diffusivity profile fits. Dashes indicate cases without a steric contribution, and “×”s indicate irreversibly deposited particles where potential energy and diffusivity profiles could not be measured. a

potentials and explain anomalous high ionic strength stability, a model was developed based on electrostatic and van der Waals potentials from the DLVO theory plus a steric repulsion attributed to silica gel layers on the particle and wall. For such a model to successfully quantify the measured potentials, the van der Waals attraction must be weakened by a layer that has some solvent composition rather than pure silica properties, although surface roughness could account for some weakening. By including an impermeable gel layer when fitting van der Waals, electrostatic, and steric potentials to measured net potentials, gel layer thicknesses of 5−10 nm were obtained from the model, consistent with literature scattering measurements. Such gel layers also indicate consistent surface separation scales for both potential energy profiles and diffusivity profiles based on theoretical models. The net potential model reported here accurately captures measured potentials and stability for [NaCl] = 0−100 mM and pH = 4−10 by including a gel layer that collapses at high ionic strengths but is relatively insensitive to pH. Our findings indicate a model of silica gel layers that captures measured van der Waals, electrostatic, steric, and hydrodynamic interactions and their role in the anomalous high ionic strength stability of silica colloids.

dependent weak acid groups could produce a dimensional collapse of the layers and an associated decreasing steric repulsion. The steric interaction indeed appears to weaken as the Debye length changes from ∼5 to 4 so that γ−1 does not change in this pH range (but might be expected to decrease as pH is lowered further). Ultimately, the results in Figure 5 in addition to the results in Figures 2−4 show potentials and stability that are well described by a silica gel layer that reduces van der Waals attraction, preserves electrostatic repulsion, and contributes an additional steric repulsion.

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CONCLUSIONS

APPENDIX Here we describe a curve fit to A(l) computed by the Lifshitz theory (eq 9). The form of the expression provides an accurate representation of both retardation and screening effects

TIRM was used to measure SiO2 colloid ensembles on a glass microscope slide to simultaneously obtain particle−wall potential energy profiles, diffusivity profiles, and stability as a function of ionic strength and pH. To interpret the measured 8842

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captured by the Lifshitz theory. This expression is convenient for computing the van der Waals potential via the Derjaguin approximation (eq 8) while avoiding the complexity and computational expense of recomputing A(l) from eq 9 each time. The form we choose is A(l) = (1/2)[1 + 2κl]exp[− 2κl]A 0 + A1 ∞(l)

(17)

where A0 is obtained from eq 9 for n = 0 (without the prefactor of 1/2(1 + 2κl)exp(−2κl) indicated by the prime symbol) and A1∞(l) is obtained from eq 9 for n = 1−∞. This form accounts for the fact that the zero frequency term (n = 0) is screened but not retarded and all higher frequency terms (n > 0) are not screened but are retarded. The function of A1∞(l) is accurately fit by A1 ∞(l) = (a f + bf l)/(1 + c f l + df l 2)

(18)

where the constants A0, af, bf, cf, and df are reported in Table 1. Figure S2, Supporting Information, shows the expression in eq 17 accurately captures of A(l) curves computed using the Lifshitz theory (eq 9).

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ASSOCIATED CONTENT

S Supporting Information *

Modeling of solution pH and ionic strength and surface potentials as well as additional figures described in the text. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge financial support by the National Science Foundation (CHE-1112335, CBET-1066254). REFERENCES

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